Question: $\overline{AB} = \sqrt{58}$ $\overline{BC} = {?}$ $A$ $C$ $B$ $\sqrt{58}$ $?$ $ \sin( \angle ABC ) = \frac{7\sqrt{58} }{58}, \cos( \angle ABC ) = \frac{3\sqrt{58} }{58}, \tan( \angle ABC ) = \dfrac{7}{3}$
Solution: $\overline{AB}$ is the hypotenuse $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle ABC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{\sqrt{58}} $ $ \overline{BC}=\sqrt{58} \cdot \cos( \angle ABC ) = \sqrt{58} \cdot \frac{3\sqrt{58} }{58} = 3$